# Duplication and Commutation

#### A Lesson in Logic

Today I’ve just got a couple of really simple substitution rules. Duplication probably won’t come up very often in regular life, but it’s necessary for a complete set. Commutation, on the other hand, is a little more common. Both of them are pretty straightforward, so we won’t spend too much time on them.

Duplication

The rule of duplication states that any proposition is equivalent to its conjunction or disjunction with itself. So P is equivalent to P ∧ P, as well as P ∨ P. Every time P is true, the others are true, so they’re interchangeable. This is a useful thing to have in your head when doing proofs, as you’ll see.

Commutation

Commutation states that any conjunction or disjunction can be reversed. So P ∧ Q can become Q ∧ P, and the same for disjunctions. It’s important to note that this rule only applies to conjunctions and disjunctions. Reversing a material conditional does change its truth values.

These are pretty easy, so this is a short post. But we’ve covered pretty much all of the easy substitution rules now. The next few can seem a little counterintuitive, but we’ll get through them. Next week will herald the return of truth tables, but before I go, here’s a logic problem for you to prove.

1. (S ∧ S) → P
2. (R ∨ P) → C
3. S ∨ S
4. Therefore, C

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